Degenerate “Star” Bifurcations in a Twinkling Oscillator

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Smruti R. Panigrahi ◽  
Brian F. Feeny ◽  
Alejandro R. Diaz
Keyword(s):  
Normal Form ◽  
Multiple Equilibria ◽  
Degree Of Freedom ◽  
Negative Stiffness ◽  
Nonlinear Spring ◽  
Mass Chain ◽  
New Type ◽  
Two Degree Of Freedom

We have studied a nonlinear spring-mass chain loaded by a quasistatic pull. The spring forces are assumed to be cubic with intervals of negative stiffness. Depending on the parameters, the system has multiple equilibria. The normal form and the bifurcation behaviors for the single- and two-degree-of-freedom systems are studied in detail. A new type of bifurcation, which we refer to as a star bifurcation, has been observed for the symmetric two-degree-of-freedom system. This bifurcation is of codimension-four for the undamped case and codimension-three or two for the damped case, depending on the form of the damping.

Bifurcations in Twinkling Oscillators

2012 ◽  
Author(s):  
Smruti R. Panigrahi ◽  
Brian F. Feeny ◽  
Alejandro R. Diaz
Keyword(s):  
High Frequency ◽  
Energy Levels ◽  
Low Frequency ◽  
Degree Of Freedom ◽  
Nonlinear Structure ◽  
High Frequency Oscillations ◽  
Mass Chain ◽  
New Type ◽  
The Masses ◽  
Two Degree Of Freedom

We present the underlying dynamics of snap-through structures that exhibit twinkling. Twinkling occurs when the nonlinear structure is loaded slowly and the masses snap-through, converting the low frequency input to high frequency oscillations. We have studied a nonlinear spring-mass chain loaded by a quasistatic pull. The spring forces are assumed to be cubic with intervals of negative stiffness. Depending on the parameters, the system has equilibria at multiple energy levels. The normal form and the bifurcation behaviors for the single and two degree of freedom systems are studied in detail. A new type of bifurcation, which we refer to as a star bifurcation, has been observed for the symmetric two degree of freedom system, which is of codimension four for the undamped case, and codimension three or two for the damped case, depending on the form of the damping.

New Type of Two Degree of Freedom Motor Three-Dimensional Tooth Layer Field Application and Solution

2013 ◽  
Vol 391 ◽  
pp. 232-236
Author(s):  
Wen Huan Yang ◽  
Hai Xu Chen ◽  
Shuang Xie ◽  
Chun Ren Fang
Keyword(s):  
Linear Equations ◽  
Three Dimensional ◽  
Field Application ◽  
Degree Of Freedom ◽  
Refined Method ◽  
New Type ◽  
Type Motor ◽  
Quasi Newton ◽  
Two Degree Of Freedom ◽  
Non Linear Equations

A new Multi-degree of freedom motor and its establishing of teeth layer parameters have been introduced in the paper, also including application method of database, namely using Quasi-Newton methods to solve the non-linear equations of the new motors magnetic circuit net, formed a refined method for designing and analyzing of motor. The establishment of 3d tooth layer parameters database, is provided for the calculation in the design of the new type motor conveniently.

Tuned mass damper and high static low dynamic stiffness isolator for vibration reduction of beam structure

2019 ◽  
Vol 234 (1) ◽  
pp. 95-115 ◽  
Author(s):  
Meysam Raei ◽  
Morteza Dardel
Keyword(s):  
Natural Frequency ◽  
Dynamic Stiffness ◽  
Excitation Amplitude ◽  
Tuned Mass Damper ◽  
Degree Of Freedom ◽  
Negative Stiffness ◽  
Low Frequencies ◽  
First Case ◽  
Mass Damper ◽  
Two Degree Of Freedom

In this work, the combination effect of tuned mass damper and high static low dynamic stiffness (HSLDS) isolator is investigated in reducing the vibration amplitude of Euler–Bernoulli beam with a nonlinear attachment. The performance of the absorber is studied in two cases; the first case, HSLDS isolator is one degree of freedom and the second case, two degree of freedom isolator is combined of HSLDS isolator and tuned mass damper absorber. By comparing the performance of these two isolators, it is revealed the two degree of freedom isolator has much better performance in direct force excitation and also improves the system performance in the base excitation. This isolator reduces the system amplitude at all frequencies, especially ultra-low frequencies, which is the main advantage to this isolator with respect to other isolators and reduces the natural frequency until the phenomenon of resonance occurs at a lower frequency. Moreover, decreasing the natural frequency increases the damping and in quasi zero stiffness and negative stiffness structure, the system has supercritical damping. This isolator is studied for positive, quasi zero and negative stiffness. The results show that the system with quasi zero stiffness has the best performances. Also, by increasing the excitation amplitude, the isolator loses its effectiveness.

scholarly journals Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2444
Author(s):  
Yani Chen ◽  
Youhua Qian
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Degree Of Freedom ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Double Hopf Bifurcation ◽  
Central Manifold ◽  
The Stability ◽  
Two Degree Of Freedom

In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV.

Manipulator Configurations Based on Rotary-Linear (R-L) Actuators and Their Direct and Inverse Kinematics

1988 ◽  
Vol 110 (4) ◽  
pp. 397-404 ◽  
Author(s):  
D. Kohli ◽  
Soo-Hun Lee ◽  
Kao-Yueh Tsai ◽  
G. N. Sandor
Keyword(s):  
Inverse Kinematics ◽  
Polynomial Equation ◽  
Open Loop ◽  
Degree Of Freedom ◽  
Serial Link ◽  
New Type ◽  
Computationally Intensive ◽  
Six Degree Of Freedom ◽  
Two Degree Of Freedom ◽  
Direct Kinematics

In this paper, a new type of two-degree-of-freedom actuator called the rotary-linear (R-L) actuator is described. The R-L actuator permits a rotation and a translation along the axis of rotation, thus simulating a cylinder pair. The R-L actuators are then used in type synthesis of mechanical manipulator chains. Closed-loop three-, four, five, and six-degree-of-freedom chains containing four to nine links, R-L actuators, revolute pairs (R), prismatic pairs (P), cylindrical pairs (C), and spheric pairs (S) are then obtained. A class of manipulator configurations where the hand is connected to the ground via six-degree-of-freedom dyads or triads and containing three grounded R-L actuators is treated for inverse kinematics. Since all the actuators are on the ground in this configuration, higher payload capacities and smaller actuator sizes can be expected from these configurations. In addition, generally, the computations required for inverse kinematics are also significantly less than those required for serial link open-loop manipulators. The direct kinematics, however, is much more involved and computationally intensive for these manipulators than for serial-link manipulators. The direct kinematics of an example manipulator is derived and requires solution of a 16th-order polynomial equation. Numerical examples are presented for illustration.

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